Localisable monads, from global to local

Abstract: Monads have many useful applications. In mathematics they are used to study algebras at the level of theories rather than specific structures. In programming languages, monads provide a convenient way to handle computational side-effects which include, roughly speaking, things like interacting with external code or altering the state of the program's variables. An important question is then how to handle several instances of such side-effects or a graded collection of them. The general approach consists in defining many “small” monads and combining them together using distributive laws.

In this talk, we take a different approach and look for a pre-existing internal structure to a monoidal category that allows us to develop a fine-graining of monads. This uses techniques from tensor topology and provides an intrinsic theory of local computational effects without needing to know how the constituent effects interact beforehand. We call the monads obtained "localisable" and show how they are equivalent to monads in a specific 2-category. To motivate the talk, we will consider two concrete applications in concurrency and quantum theory. This is all covered in our recent paper: arxiv.org/abs/2108.01756 .

category theory

Audience: researchers in the topic

( paper )

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